Materials for teaching the axioms of the real numbers to high school students I suddenly felt the urge to teach the axioms of the real numbers (i.e. the complete ordered field axioms) to a bright Year 10 student that I tutor, with an emphasis on the consequences of the field axioms. Unfortunately, I haven't been able to find any good materials (expository pieces, worksheets, etc.) to help me do this. Recommendations, anyone?
 A: This sounds like one of those things easiest to accomplish by visiting a university library and browsing the shelves at the locations of real analysis books, transition to advanced mathematics type books, and books dealing with constructions and axiomatizations of the real numbers. I say this because you and your student's background and interests will play a huge role in what is appropriate, and the quickest way of weeding through the large amount of material available is to be in a position where you can immediately look at something for consideration.
Off the top of my head, the following book (which fits into the 3rd category I listed) seems like it might be a good fit:
Stefan Drobot, Real Numbers, Prentice-Hall, 1964, x + 102 pages.

(My comments about the book) This book begins with an axiomatic treatment of the real numbers as a complete ordered field and then discusses various expansions of real numbers (decimal, Cantor, continued fraction), approximation of irrationals by rationals, and (briefly) cardinality and measure of sets of real numbers.

Below are some other possible books. Incidentally, there were a lot of books published in the U.S. during the 1960s having to do with constructions and axiomatizations of the real numbers.
Leon Warren Cohen and Gertrude Ehrlich, The Structure of the Real Number System, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, 1963, viii + 116 pages.
Solomon Feferman, The Number Systems. Foundations of Algebra and Analysis, Addison-Wesley Publishing Company, 1964, xii + 418 pages.
Norman Tyson Hamilton and Joseph Landin, Set Theory and the Structure of Arithmetic, Allyn and Bacon, 1961, xii + 264 pages.
Edmund Jecheksel Landau, Foundations of Analysis, 1951, Chelsea Publishing Company, 1951, xiv + 134 pages.
Elliott Mendelson, Number Systems and the Foundations of Analysis, Academic Press, 1973, xii + 358 pages.
John Meigs Hubbell Olmsted, The Real Number System, Appleton-Century Monographs in Mathematics, Appleton-Century-Crofts, 1962, xii + 216 pages.
Francis Dunbar Parker, The Structure of Number Systems, Teachers' Mathematics Reference Series, Prentice-Hall, 1966, xiv + 137 pages.
Joseph [Joe] Buffington Roberts, The Real Number System in an Algebraic Setting, A Series of Undergraduate Books in Mathematics, W. H. Freeman and Company, 1962, x + 145 pages.
A: High school students might be able to learn how to formally deduce properties of the real numbers from those axioms, but for some students, the proof of its structure might not be very convincing. I was introduced to the concept of infinity early outside of school so I formed the intuition that $\infty$ exceeds all natural numbers and $\infty + 1 \neq \infty$ and $1 \div \infty \neq 0$. Those who already formed the intuition of the hyperreal number system might reject the claim of completeness of the real number system. They might be like "I know that once you fill the Dedekind cuts of the [hyperreal] number system and then invent all the numbers that can be gotten from them using the operation of addition and multiplication, it completes only the original system but the new system is still incomplete. That doesn't mean the infinitesimals don't even exist to start with. It just means that there exists even more numbers outside of the new system. For any ordinal number no matter how large, you can complete the system that number of times. Now take the system of all numbers that can be gotten by completing the hyperreal number system any ordinal number of times. Even this system is incomplete. However, there are no numbers that fill the holes in this system. This can be explained by rejecting Naive set theory. You cannot invent a number to fill a hole of that system when you have not already invented all the ordinal numbers that exist." I think it might instead be better to give an explicit construction of real numbers and their operations as follows.
First we construct the natural numbers as finite ordinal numbers and define +, $\times$, and $\leq$ on them. Next we construct the integers as follows. There is no solution to $x + 1 = 0$ so we invent a solution -1. There is still no solution to $x + 1 = -1$ so we invent a solution -2. Next we construct the dyadic rationals, the numbers with a terminating notation in base 2 as follows. Each odd number $x$ is not a solution to $2 \times y = x$ in $\mathbb{Z}$ so we invent a solution to it. Let's call each newly invented number a half integer. Each half integer $y$ is still not a solution to $2 \times z = y$ so we invent a solution for it. We keep going for ever and then again redefine +, $\times$, and $\leq$ on them.
I know this is not the definition of a Dedekind cut but I will define a Dedekind cut of the dyadic rationals to be a subset of the dyadic rationals such that it is nonempty and its complement is nonempty and for every dyadic rational in the subset, all smaller dyadic rationals are in the subset. Some Dedekind cuts have a maximal element. Some Dedekind cuts have a complement with a minimal element. Some Dedekind cuts have no maximal element nor does their complement have a minimal element. For each Dedekind cut that has no maximal element nor does its complement have a minimal element, we can invent a number that's larger than all elements of the Dedekind cut and smaller than all elements of its complement to get the real numbers. Again, we can redefine +, $\times$, and $\leq$ on them in the intuitive way and show that it is a complete ordered field which is unique up to isomorphism.
Maybe even for those who formed the intuition of the existence of infinitesimal numbers, they can be taught that this is the definition of the real number system and in that system, there are no infinitesimal numbers.
